Statistical Inference

Definition 77

The Maximum A Posteriori (MAP) estimate is given by

Definition 78

The Maximum Likelihood (ML) estimate is given by

Binary Hypothesis Testing

Definition 79

Definition 80

Definition 81

With two possible hypotheses, there are two kinds of errors we can make.

Definition 82

A Type I error (false positive) is when we incorrectly reject the null hypothesis. The Type I error probability is then

Definition 83

A Type II error (false negative) is when we incorrectly accept the null hypothesis. The Type II error probability is then

Definition 84

The likelihood ratio is given by

Definition 85

MAP fits into the framework of a threshold test since we can write

It turns out that threshold tests are optimal with respect to solving Equation 1.

Theorem 44 (Neyman Pearson Lemma)

is a threshold test.

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